The best place to get a camera is probably the camera store. ^^
Anyways, as jowens said many of us are self-taught. I myself haven't had a single course in photography or the likes, I learnt everything through practice and the internet (this forum helps loads for example).
There is no single answer when it comes to best equipment. Each and every brand has its own advantages and perks.
The more popular ones are Nikon and Canon. Sony is also a rather popular brand, although I have no knowledge of that brand at all.
I myself shoot Nikon, the main reasons for this being that I liked the menus more and most of the photographers I know in real life shoot Nikon. This has the added advantage that we can easily borrow each others lenses.
I suggest you go to your local camera store and ask if you can hold a few camera's just to feel them. Pick the one that feels nicest, you're going to be holding it a lot.
Assuming you want a DSLR camera you're probably looking at a D3100/D3200 for Nikon, these are very nice beginner models and capable of making really good photos. I'm not sure what the Canon/Sony alternative for these models is... (Someone help me please

).
Of course other models are probably good too. If you're willing to spend more you can go for some more advanced models (I started with a D7000 myself). It's all up to you.
Yes, gong outside and taking photos is generally speaking a rather effective way of learning photography. Practice makes perfect.
Your fiances dogs will probably make a fun subject.
Besides that you're probably going to want to read up on exposure and composition to enlarge your knowledge (which you can then practice with the dogs). There are numerous of articles on the internet on these subjects.